Finite Group Databasebeta

← #13159417

Subgroups — #13159417

Order 1728.

1388 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1388 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
2C3×#64104C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592472—31nomaximal
9#10590595—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×D12C2211yescharacteristic
16C2×#6291—21no—
17C3⋊D3×D12—22no—
18C3⋊D3×D12—22no—
19C33×C4⋊C4C2211yescharacteristic, complemented
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C3×#3752—21no—
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88791—22no—
24#88789—22no—
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—34no—
27C2×#3753—31no—
28C2×#3745—31no—
29#64104D311yescharacteristic, complemented
30C3×#895—34no—
31C2×#3714—34no—
32C2×#3713—34no—
33C32×#102—31no—
34#63647—31no—
35#63636—31no—
36#63636—34no—
37#63636—64no—
38C3×#809D314yescomplemented
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—21no—
44C2×D3×C3⋊D3—24no—
45C2×D3×C3⋊D3—42no—
46C2×D3×C3⋊D3—42no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))D412yescomplemented
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—42no—
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C32×D12—21no—
58C32×D12D412yescomplemented
59#6291—24no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C32×C3⋊C8—21no—
62#5036—91no—
63#4773—94no—
64C23×C3⋊D3—61no—
65C22×D32—64no—
66C2×C32×D4—31no—
67C2×#638—61no—
68D4×C3⋊D3—62no—
69D4×C3⋊D3—62no—
70C2×C4⋊(C3⋊D3)—21no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—34no—
73C2×C4⋊(C3⋊D3)—64no—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×D12—34no—
77C2×C3×D12—64no—
78C2×C3×D12C2×D314yes—
79C2×C3×C4×D3—34no—
80C2×#626—64no—
81D3×D12—68no—
82D3×D12—68no—
83C2×C32×C8—31no—
84C32×C4⋊C4—24no—
85C32×C4⋊C4C2×D314yescomplemented
86C32×C4⋊C4C2×D311yescharacteristic, complemented
87#3753—62no—
88#3752—21no—
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—34no—
91C2×C3⋊(C3⋊C8)—64no—
92#3745—62no—
93C3×(C2×D3)⋊C4—64no—
94C2×C3×C3⋊C8C2×D314yescomplemented
95#3714—68no—
96#3713—68no—
97C22×C33C2×D411yescharacteristic
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3D812yescomplemented
107C2×C32×D3—22no—
108C2×C32×D3—41no—
109C2×C32×D3—41no—
110D3×C3⋊D3—48no—
111C33×C4—41no—
112C33×C4C2×D411yescharacteristic
113C33×C4C2×D411yescharacteristic
114C3×C3⋊Dic3—21no—
115C2×D3×D4—94no—
116C22×D12—91no—
117C3×#121—91no—
118C2×#213—94no—
119C2×#213—91no—
120C2×#213—91no—
121C2×D24—94no—
122#895—34no—
123C3×#102—34no—
124#830—91no—
125(C2×D3)⋊C8—94no—
126#809—94no—
127#809—91no—
128#809—184no—
129#809C3⋊D311yescharacteristic, complemented
130C23×C32—61no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—64no—
135C22×C3⋊D3—124no—
136C22×C3⋊D3—122no—
137C22×C3⋊D3—122no—
138C22×C3⋊D3#6711yescharacteristic
139C22×C3×D3—24no—
140C22×C3×D3—34no—
141C22×C3×D3—64no—
142C2×D32—616no—
143C2×D32—128no—
144C2×D32—128no—
145C32×D4—32no—
146C32×D4—61no—
147C2×C32×C4—21no—
148C2×C32×C4—24no—
149C2×C32×C4—24no—
150C2×C32×C4—44no—
151C2×C32×C4C22×D314yes—
152C2×C32×C4C22×D311yescharacteristic
153#638—64no—
154C2×C3⋊Dic3#6711yescharacteristic
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)—32no—
158C4⋊(C3⋊D3)—38no—
159C4⋊(C3⋊D3)—42no—
160C4⋊(C3⋊D3)—68no—
161C4⋊(C3⋊D3)—61no—
162C4⋊(C3⋊D3)—64no—
163C4⋊(C3⋊D3)—124no—
164C4×C3⋊D3—21no—
165C4×C3⋊D3—21no—
166C2×C3×Dic3—34no—
167C3×D12—24no—
168C3×D12—64no—
169C3×D12—64no—
170C3×D12—128no—
171C3×D12D1218yescomplemented
172C3×C4×D3—64no—
173C3×C4×D3—64no—
174#626—616no—
175C32×C8—61no—
176C3⋊(C3⋊C8)—61no—
177C3⋊(C3⋊C8)—64no—
178C3⋊(C3⋊C8)—124no—
179C3×C3⋊C8—24no—
180#447—271noSylow
181C2×C33#12011yescharacteristic
182C2×C33C2×D811yescharacteristic
183C2×C33AGL(1,ℤ/8)11yescharacteristic
184C3⋊(C3⋊D3)—42no—
185C3×C3⋊D3—22no—
186C3×C3⋊D3—81no—
187C32×D3—42no—
188C23×D3—181no—
189C23×D3—184no—
190C2×C3×D4—34no—
191C2×C3×D4—91no—
192C2×C3×D4—181no—
193C22×C3×C4—91no—
194C2×#67—184no—
195D3×D4—188no—
196D3×D4—188no—
197C2×D12—34no—
198C2×D12—64no—
199C2×D12—94no—
200C2×D12—91no—
201C2×D12—184no—
202C2×D12—182no—
203C2×D12—181no—
204C2×D12—182no—
205C2×D12C2×C3⋊D311yescharacteristic
206C2×C4×D3—34no—
207C2×C3×C8—34no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4D3214yescomplemented
210C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
211C3×C22⋊C4—181no—
212#213—182no—
213#213—188no—
214#213—182no—
215(C2×D3)⋊C4—64no—
216C2×C3⋊C8—94no—
217C2×C3⋊C8—91no—
218C2×C3⋊C8—184no—
219C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
220D24—188no—
221C22×C32—24no—
222C22×C32—62no—
223C22×C32—121no—
224C22×C32—121no—
225C22×C32D3×D414yes—
226C22×C32C2×#6711yescharacteristic
227C2×C3⋊D3—21no—
228C2×C3⋊D3—21no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3—41no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—62no—
233C2×C3⋊D3—68no—
234C2×C3⋊D3—128no—
235C2×C3⋊D3—128no—
236C2×C3⋊D3—121no—
237C2×C3⋊D3—121no—
238C2×C3⋊D3—124no—
239C2×C3⋊D3—124no—
240C2×C3⋊D3—244no—
241C2×C3⋊D3—244no—
242C2×C3⋊D3#21312yescomplemented
243C2×C3×D3—28no—
244C2×C3×D3—38no—
245C2×C3×D3—44no—
246C2×C3×D3—44no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3—124no—
252D32—1232no—
253C32×C4—24no—
254C32×C4—24no—
255C32×C4—41no—
256C32×C4—44no—
257C32×C4—84no—
258C32×C4C2×D1214yes—
259C32×C4D3×D414yes—
260C32×C4D3×D411yescharacteristic
261C32×C4D3×D411yescharacteristic
262C3⋊Dic3—21no—
263C3×Dic3—64no—
264C22×D4—271no—
265C2×D8—271no—
266C2×D8—271no—
267#121—91no—
268#102—31no—
269#102—271no—
270C22⋊C8—271no—
271C33#44711yescharacteristic, complemented, Sylow
272C23×C3—64no—
273C23×C3—91no—
274C23×C3—181no—
275C22×D3—21no—
276C22×D3—34no—
277C22×D3—64no—
278C22×D3—184no—
279C22×D3—1816no—
280C22×D3—184no—
281C22×D3—181no—
282C22×D3—364no—
283C22×D3—368no—
284C22×D3—362no—
285C22×D3—362no—
286C22×D3—368no—
287C3×D4—38no—
288C3×D4—64no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4—362no—
292C2×C3×C4—24no—
293C2×C3×C4—21no—
294C2×C3×C4—24no—
295C2×C3×C4—44no—
296C2×C3×C4—91no—
297C2×C3×C4—181no—
298C2×C3×C4—181no—
299C2×C3×C4C2×D3214yes—
300C2×C3×C4C22×C3⋊D311yescharacteristic
301#67—1816no—
302C2×Dic3—34no—
303D12—21no—
304D12—64no—
305D12—64no—
306D12—98no—
307D12—92no—
308D12—128no—
309D12—188no—
310D12—184no—
311D12—184no—
312D12—181no—
313D12—364no—
314D12C4⋊(C3⋊D3)12yescomplemented
315C4×D3—64no—
316C4×D3—64no—
317C3×C8—64no—
318C3⋊C8—21no—
319C3⋊C8—184no—
320C3⋊C8—181no—
321C3⋊C8—364no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—122no—
326C2×C32#88214yes—
327C2×C32D3×D814yes—
328C2×C32#91414yes—
329C2×C32C2×#21311yescharacteristic
330C2×C32#93711yescharacteristic
331C2×C32#94911yescharacteristic
332C3⋊D3—22no—
333C3⋊D3—81no—
334C3⋊D3—122no—
335C3⋊D3—128no—
336C3⋊D3—248no—
337C3×D3—48no—
338C3×D3—68no—
339C3×D3—244no—
340C24—541no—
341C2×D4—31no—
342C2×D4—91no—
343C2×D4—181no—
344C2×D4—271no—
345C2×D4—541no—
346C2×D4—542no—
347C2×D4—542no—
348C22×C4—91no—
349D8—542no—
350D8—542no—
351C2×C8—31no—
352C2×C8—271no—
353C4⋊C4D3×C3⋊D311yescharacteristic, complemented
354C22⋊C4—181no—
355C22×C3—24no—
356C22×C3—68no—
357C22×C3—92no—
358C22×C3—124no—
359C22×C3—124no—
360C22×C3—181no—
361C22×C3—181no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3D3×#6714yes—
366C22×C3D4×C3⋊D311yescharacteristic
367C2×D3—22no—
368C2×D3—38no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—64no—
372C2×D3—64no—
373C2×D3—124no—
374C2×D3—124no—
375C2×D3—124no—
376C2×D3—182no—
377C2×D3—188no—
378C2×D3—368no—
379C2×D3—368no—
380C2×D3—3632no—
381C2×D3—361no—
382C2×D3—361no—
383C2×D3—364no—
384C2×D3—364no—
385C2×D3—724no—
386C2×D3—724no—
387C3×C4—24no—
388C3×C4—24no—
389C3×C4—41no—
390C3×C4—44no—
391C3×C4—84no—
392C3×C4—181no—
393C3×C4D3×D1214yes—
394C3×C4#380214yes—
395C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
396C3×C4D4×C3⋊D311yescharacteristic
397Dic3—64no—
398C32—24no—
399C32#477314yescomplemented
400C32#503611yescharacteristic, complemented
401C23—61no—
402C23—91no—
403C23—181no—
404C23—544no—
405C23—541no—
406C23—1082no—
407C23—1082no—
408D4—32no—
409D4—61no—
410D4—181no—
411D4—181no—
412D4—272no—
413D4—362no—
414D4—544no—
415D4—541no—
416C2×C4—21no—
417C2×C4—91no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C8—61no—
422C8—541no—
423C2×C3—24no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—128no—
427C2×C3—182no—
428C2×C3—721no—
429C2×C3#6392214yes—
430C2×C3D3×#21314yes—
431C2×C3#6393914yes—
432C2×C3#6409111yescharacteristic
433C2×C3D8×C3⋊D311yescharacteristic
434C2×C3#6412311yescharacteristic
435D3—42no—
436D3—68no—
437D3—244no—
438D3—368no—
439D3—362no—
440D3—728no—
441C22—62no—
442C22—92no—
443C22—121no—
444C22—121no—
445C22—181no—
446C22—181no—
447C22—361no—
448C22—361no—
449C22—361no—
450C22—542no—
451C22—1088no—
452C22—1081no—
453C22—1081no—
454C22C3⋊D3×#6711yescharacteristic
455C4—41no—
456C4—181no—
457C4D3×C4⋊(C3⋊D3)11yescharacteristic
458C4#8902611yescharacteristic
459C3—24no—
460C3#1059059514yescomplemented
461C3#1059247211yescharacteristic, complemented
462C2—122no—
463C2—182no—
464C2—721no—
465C2—1082no—
466C2#1172354111yescharacteristic
467C2C3⋊D3×#21311yescharacteristic
468C2#1172357011yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#447
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3⋊D3×D12C2semidirect
2. C3×#64104C2semidirect
3. C2×#88791C2semidirect
4. C2×#88789C2semidirect
5. #11722179C2semidirect
6. C32×#809C2semidirect
7. #11722023C2semidirect
13. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C33×C4⋊C4C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. C2×C32×C3⋊C8C22semidirect
29. #64104D3semidirect
38. C3×#809D3semidirect
50. C4⋊(C3⋊(C3⋊D3))D4semidirect
58. C32×D12D4semidirect
74. C2×C4⋊(C3⋊D3)C2×D3semidirect
85. C32×C4⋊C4C2×D3semidirect
86. C32×C4⋊C4C2×D3semidirect
94. C2×C3×C3⋊C8C2×D3semidirect
106. C2×C3×C3⋊D3D8semidirect
129. #809C3⋊D3semidirect
171. C3×D12D12semidirect
209. C3×C4⋊C4D32semidirect
210. C3×C4⋊C4C2×C3⋊D3semidirect
219. C2×C3⋊C8C2×C3⋊D3semidirect
242. C2×C3⋊D3#213semidirect
271. C33#447semidirect
314. D12C4⋊(C3⋊D3)semidirect
353. C4⋊C4D3×C3⋊D3semidirect
399. C32#4773semidirect
400. C32#5036semidirect
460. C3#10590595semidirect
461. C3#10592472semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3⋊D3×D122
C3×#641042
C2×#887912
C2×#887892
#117221792
C32×#8092
#117220232
#105924723
#105905953